On mixed quasi-Einstein spacetimes

Abstract

The object of the present paper is to study mixed quasi-Einstein spacetimes, briefly $M(QE)_4$ spacetimes. First we prove that every $Z$ Ricci pseudosymmetric $M(QE)_4$ spacetimes is a $Z$ Ricci semisymmetric spacetimes. Then we study $Z$ flat spacetimes. Also we consider Ricci symmetric $M(QE)_4$ spacetimes and among others we prove that the local cosmological structure of a  Ricci symmetric $M(QE)_{4}$ perfect fluid spacetime can be identified as Petrov type $I$, $D$ or $O$.  We show that such a spacetime is the Robertson-Walker spacetime. Moreover we deal with mixed quasi-Einstein spacetimes with the associated generators $U$ and $V$ as concurrent vector fields. As a consequence we obtain some important theorems. Among others it is shown that a perfect fluid $M(QE)_4$ spacetime of non zero scalar curvature with the basic vector field of spacetime as velocity vector field of the fluid is of Segr\'{e} characteristic $[(1,1,1),1]$. Also we prove that a $M(QE)_{4}$ spacetime can not admit heat flux provided the smooth function $b$ is not equal to the cosmological constant $k$. This means that such a spacetime describe a universe which has already attained thermal equilibrium. Finally, we construct a non-trivial Lorentzian metric of $M(QE)_4$.